Abstract
Schnorr randomness and computably randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness (defined by Franklin and Stephan too only by martingales) and truth-table reducible randomness, for which we prove that van Lambalgen’s Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen’s Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness.